Anisotropic Maximal $L^p$-regularity Estimates for a Hypoelliptic Operator

We consider the maximal regularity of a specific Vlasov-Fokker-Planck equation $\mathcal{A}u=f$ in the Euclidean space. The operator $\mathcal{A}={\mathrm{\Delta }}_{y}u-y\cdot {\mathrm{\nabla }}_{x}u$ is an example of the Ornstein-Uhlenbeck operators. We prove the existence of a solution that satisfies the anisotropic maximal regularity estimates. To prove this we also show a similar estimates and a weak (1, 1) estimate for $L={\mathrm{\partial }}_t-\mathcal{A}$, which is of independent interest. These results rely on the pointwise estimates of the fundamental solution of $L$.